
# 7.3: Unusual Number systems


## Dualtown Number System

The peoples of Dualtown use only numbers which are 1 and some of the multiples of 2 (even numbers).   $$E=\{1, 2, 4, 6,8 \cdots\}$$. Notice that the set $$E$$ is closed under multiplication.

Let's construct a $$4 \times 4$$ multiplication table $$E$$.

 $$\times$$ $$1$$ $$2$$ $$4$$ $$6$$ $$1$$ $$1$$ $$2$$ $$4$$ $$6$$ $$2$$ $$2$$ $$4$$ $$8$$ $$12$$ $$4$$ $$4$$ $$8$$ $$16$$ $$24$$ $$6$$ $$6$$ $$12$$ $$24$$ $$36$$

The smallest ten prime numbers in Dualtown are $$2, 6, 10, 14, 18, 22, 26, 30, 34, 38$$. Notice that $$36=(6)(6)=(2)(18)$$. Thus $$36$$ has two different Dualtown prime factorizations.  Hence the Prime Divisibility Theorem  does not hold for the Dualtown number system.